Certainly! Let's determine the equation of a circle given its center and radius.
The standard form of the equation of a circle is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
Given:
- The center of the circle is [tex]\((3, 7)\)[/tex], so [tex]\(h = 3\)[/tex] and [tex]\(k = 7\)[/tex].
- The radius [tex]\(r = 4\)[/tex].
Substitute these values into the standard form equation:
1. Substitute [tex]\(h\)[/tex] and [tex]\(k\)[/tex] with 3 and 7, respectively:
[tex]\[
(x - 3)^2 + (y - 7)^2 = r^2
\][/tex]
2. Substitute [tex]\(r\)[/tex] with 4:
[tex]\[
(x - 3)^2 + (y - 7)^2 = 4^2
\][/tex]
3. Simplify the right side of the equation:
[tex]\[
4^2 = 16
\][/tex]
So the equation becomes:
[tex]\[
(x - 3)^2 + (y - 7)^2 = 16
\][/tex]
Therefore, the equation that describes the circle with center [tex]\((3, 7)\)[/tex] and radius 4 is:
[tex]\[
(x - 3)^2 + (y - 7)^2 = 16
\][/tex]
Let's compare this with the given options:
A. [tex]\((x + 3)^2 + (y + 7)^2 = 16\)[/tex] [tex]\(\quad\)[/tex] (Incorrect form)
B. [tex]\((x - 3)^2 + (y - 7)^2 = 4\)[/tex] [tex]\(\quad\)[/tex] (Incorrect radius)
C. [tex]\((x - 3)^2 + (y - 7)^2 = 16\)[/tex] [tex]\(\quad\)[/tex] (Correct answer)
D. [tex]\((x + 3)^2 + (y + 7)^2 = 4\)[/tex] [tex]\(\quad\)[/tex] (Incorrect form and radius)
The correct answer is [tex]\( \boxed{3} \)[/tex].
